A model of parallel optical computer for Fourier transform tasks is built. The computer operates in both quantum and classical modes. The quantum computer (QC) works with entangled photons and the input gates are n-qubit sensitive, while the classical computer (CC) operates without entanglement. The information capacity (I) for both CC and QC is given as a function of number of qubits and Np (number of photons per input pixel). It is shown that the information capacity significantly increases for the quantum computer compared to the classical one as the number of qubits is increased, assuming that Np is constant for both the QC and CC, i.e. same amount of energy input is assumed for the computers. However, it is also pointed out that the complexity of the QC significantly increases, too. To quantify this, we introduce a new physical quantity called physical complexity (noted as Q). We define the physical complexity (Q) of a computer as ∑i,klog2nkai where k runs over all the gates/elements in the computer; nai is the number of distinguishable states what a gate can set for ai. For QC ai is defined as the ith coefficient of the wave function. For CC it is the ith independent component of the Fourier transform of the energy flow what a particular gate is able to control. By using this definition we show that, for both the quantum and classical computers built in this work, I ≤ Q, in other words the information capacity is less or equal than the physical complexity. Intuitively we suggest that this is a general law that is valid for every computer irrespectively of type, quantum or classic, and architecture.
A model of parallel optical computer for Fourier transform tasks is built. The computer operates in both quantum and classical modes. The quantum computer (QC) works with entangled photons and the input gates are n-qubit sensitive, while the classical computer (CC) operates without entanglement. The information capacity (I) for both CC and QC is given as a function of number of qubits and Np (number of photons per input pixel). It is shown that the information capacity significantly increases for the quantum computer compared to the classical one as the number of qubits is increased, assuming that Np is constant for both the QC and CC, i.e. same amount of energy input is assumed for the computers. However, it is also pointed out that the complexity of the QC significantly increases, too. To quantify this, we introduce a new physical quantity called physical complexity (noted as Q). We define the physical complexity (Q) of a computer as ∑i,klog2nkai where k runs over all the gates/elements in the computer; nai is the number of distinguishable states what a gate can set for ai. For QC ai is defined as the ith coefficient of the wave function. For CC it is the ith independent component of the Fourier transform of the energy flow what a particular gate is able to control. By using this definition we show that, for both the quantum and classical computers built in this work, I ≤ Q, in other words the information capacity is less or equal than the physical complexity. Intuitively we suggest that this is a general law that is valid for every computer irrespectively of type, quantum or classic, and architecture.
A gedanken model of an analog optical computer for general purpose Fourier transform tasks is built and studied assuming entangled or non-entangled photons appearing at the input. The performance of the entangled and non-entangled operation modes, regarding information content, power consumption, acquisition time, wavelength and bandwidth limitations are analyzed as function of qubit repetition rate and number of qubits in the entangled state, respectively, based on fundamental physical laws. The total number of input photons is assumed the same for the two operation modes. It is sown that the entangled mode outperforms the non-entangled one, but only above a certain number of qubits in the entangled state. It is argued that, for practical applications to general purpose computing, several limitations should be imposed regarding qubit repetition rate and number of qubits in the entangled state. In addition, it is pointed out that the conclusions that we present are equally valid for general purpose optical quantum computing and for quantum information processing, too.
AbstractThe quantum computing has been proven to provide the solutions to most complex computations in the world today. Quantum computers can solve the problems with polynomial resources. Quantum information can be encoded in a number of ways, such as spin components of basic particles like photons, electrons, and polarization of photons. The quantum computer uses quantum bits which are superposition of states called as qubits, which is the classical fundamental unit of information. A quantum computer is a regular classical computer with additional ability to control the quantum bits. It is a regular computer connected to a special peripherals and a set of quantum bits simulated to build controller architecture. Two simulators are used for the quantum programming, one is Microsoft Q# and other is IBM Qiskit. Classical language Q # is used to program the quantum controller and its operations such that initializing a qubit, applying a gate on a qubit and measuring a qubit. IBM Qiskit programs simulates the functionality of quantum circuit and creates it’s drawing using a graphical interface with qubits, gates, and measurements. Qiskit provides simulators that run quantum circuits using visualization tools that help to understand the effect of our quantum circuits on the quantum states. The teleportation technique allows the quantum information to be communicated at a distance. An unbreakable encryption protocol is built for polarizing photons through the quantum channel. The combination of quantum computing and AI has the potential to drive significant advances across many sectors.
This Spotlight aims to provide a general introduction to linear and nonlinear optical components that are frequently used to implement the protocols for quantum computation and communication. The role of each optical element is described briefly, and it is shown that these optical elements can be combined to constitute quantum circuits for performing various quantum computing and communication tasks. Present challenges and future scopes are also mentioned briefly. Specifically, after introducing the qubit, it is stated that a quantum gate is a device that transforms a quantum state into another quantum state. Thus, most of the optical components (e.g., a beamsplitter, a wave plate, a mirror, a polarizing beamsplitter) can be viewed as quantum gates because each of these optical elements can transform a quantum state into another quantum state. This fact is the basis for constructing useful quantum circuits using optical elements. Finally, a few proposals for optical realizations of schemes for quantum communication and computation are described.
In classical computation and communication, we use classical gates and bits. Similarly, in quantum computation and communication, we use their quantum analogs, i.e., quantum gates and qubit(s) [quantum bit(s)]. We have already mentioned that a bit can be realized in various ways. In the same way, a qubit can also be realized in various ways, and accordingly, corresponding quantum gates that would operate on qubits can also be realized in various ways. Here, we focus on a specific implementation: optical implementation, where qubits are realized using photons and quantum gates, and circuits are realized using various optical components. Specific merits of this realization lie in the following facts: (i) in quantum communication we need to transmit qubits from one location to the other, and transmission of photons is easier than any other realization of qubits; (ii) the effect of the environment (e.g., external electric and magnetic fields) on photons is less compared to any other implementations of qubits; and (iii) integrated optics are already a well-developed field. Keeping these points in mind, this Spotlight describes basic optical elements and how to use those elements to implement various schemes of quantum computation and communication. We will mostly describe realizations using bulk optics. However, in most cases, it is not difficult to transform these bulk-optics-based realizations into integrated optics. To begin with, in Section 1.1, we formally define a qubit and describe two alternative ways of realizing a qubit using a photon.
We give new evidence that quantum computers -- moreover, rudimentary quantum computers built entirely out of linear-optical elements -- cannot be efficiently simulated by classical computers. In particular, we define a model of computation in which identical photons are generated, sent through a linear-optical network, then nonadaptively measured to count the number of photons in each mode. This model is not known or believed to be universal for quantum computation, and indeed, we discuss the prospects for realizing the model using current technology. On the other hand, we prove that the model is able to solve sampling problems and search problems that are classically intractable under plausible assumptions. Our first result says that, if there exists a polynomial-time classical algorithm that samples from the same probability distribution as a linear-optical network, then P^#P=BPP^NP, and hence the polynomial hierarchy collapses to the third level. Unfortunately, this result assumes an extremely accurate simulation. Our main result suggests that even an approximate or noisy classical simulation would already imply a collapse of the polynomial hierarchy. For this, we need two unproven conjectures: the "Permanent-of-Gaussians Conjecture", which says that it is #P-hard to approximate the permanent of a matrix A of independent N(0,1) Gaussian entries, with high probability over A; and the "Permanent Anti-Concentration Conjecture", which says that |Per(A)|>=sqrt(n!)/poly(n) with high probability over A. We present evidence for these conjectures, both of which seem interesting even apart from our application. This paper does not assume knowledge of quantum optics. Indeed, part of its goal is to develop the beautiful theory of noninteracting bosons underlying our model, and its connection to the permanent function, in a self-contained way accessible to theoretical computer scientists.
We give new evidence that quantum computers -- moreover, rudimentary quantum computers built entirely out of linear-optical elements -- cannot be efficiently simulated by classical computers. In particular, we define a model of computation in which identical photons are generated, sent through a linear-optical network, then nonadaptively measured to count the number of photons in each mode. This model is not known or believed to be universal for quantum computation, and indeed, we discuss the prospects for realizing the model using current technology. On the other hand, we prove that the model is able to solve sampling problems and search problems that are classically intractable under plausible assumptions. Our first result says that, if there exists a polynomial-time classical algorithm that samples from the same probability distribution as a linear-optical network, then P#P=BPPNP, and hence the polynomial hierarchy collapses to the third level. Unfortunately, this result assumes an extremely accurate simulation.
We have seen in the previous chapters that quantum computers seem to be more powerful than classical computers for certain problems. There are limits on the power of quantum computers, however. Since a classical computer can simulate a quantum computer, a quantum computer can only compute the same set of functions that a classical computer can. The advantage of using a quantum computer is that the amount of resources needed by a quantum algorithm might be much less than what is needed by the best classical algorithm. In Section 9.1 we briefly define some classical and quantum complexity classes and give some relationships between them. Most of the interesting questions relating classical and quantum complexity classes remain open. For example, we do not yet know if a quantum computer is capable of efficiently solving an NP-complete problem (defined later). One can prove upper bounds on the difficulty of a problem by providing an algorithm that solves that problem, and proving that it will work within in a given running time. But how does one prove a lower bound on the computational complexity of a problem? For example, if we wish to find the product of two n-bit numbers, computing the answer requires outputting roughly 2n bits and that requires Ω(n) steps (in any computing model with finite-sized gates). The best-known upper bound for integer multiplication is O(n log n log log n) steps. It has proved extremely difficult to derive non-trivial lower bounds on the computational complexity of a problem. Most of the known non-trivial lower bounds are in the ‘black-box’ model (for both classical and quantum computing), where we only query the input via a ‘black-box’ of a specific form. We discuss the black-box model in more detail in Section 9.2. We then sketch several approaches for proving black-box lower bounds. The first technique has been called the ‘hybrid method’ and was used to prove that quantum searching requires Ω(√n) queries to succeed with constant probability. The second technique is called the ‘polynomial method’. We then describe a technique based on ‘block sensitivity’, and conclude with a technique known as the ‘adversary method’.